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A uniform mass density slab with free surfaces at $z=\pm H$ has ${\rho }_0=\Sigma /(2H)$, internal gravity $4\pi G{\rho }_{0}z$ and hydrostatic pressure $2\pi G{\rho }_0^2(H^2-z^2)$ above ambient pressure. Its surface modes of a self-gravitating incompressible slab separate into thickness and bending parities.

For a constant-luminosity radiative envelope with ideal-gas pressure and Kramers' opacity law, impose power laws on enclosed mass, density, pressure and temperature while retaining $M_r^{\prime }=4\pi r^2\rho$. If their exponents are $m,b,p,d$, respectively, mass conservation gives $m=b+3$, hydrostatic balance gives $d=m-1$, the gas law gives $d=p-b$, and diffusion gives $d-1=2b-2-13d/2$. Solving gives $(m,b,p,d)=(1,-32,-42,-10)/11$. In particular $R/r_c=(T_c/T_R{)}^{11/10}$. Neglecting envelope self-gravity instead sets $M_r$ approximately constant and changes the exponents to $d=-1,b=-13/4,p=-17/4$; that approximation is not an exact solution of mass conservation with nonzero density.

Self-gravity is the gravitational acceleration produced by a system's own mass density. It is obtained by solving the Poisson equation for Newtonian gravity for that density, rather than treating the Newtonian gravitational potential as an externally fixed field. It competes with pressure and rotational support in gravitational instability of an astrophysical disk.

For $\gamma >1$ and positive mass density and pressure, let $\xi =x/t$ in one-dimensional adiabatic compressible flow. The smooth equations are $(u-\xi ){\rho }^{\prime }+\rho u^{\prime }=0$, $(u-\xi )u^{\prime }+p^{\prime }/\rho =0$ and $(u-\xi )p^{\prime }+\gamma p{u}^{\prime }=0$. Away from a constant branch, their coefficient determinant forces $\xi =u+\sigma c$ with $\sigma =\pm 1$ and adiabatic sound speed $c=\sqrt{\gamma p/\rho }$. The entropy equation gives $p=K{\rho }^{\gamma }$, and combining continuity with $c^{\prime }/c=(\gamma -1){\rho }^{\prime }/(2\rho )$ givesThus $u=[2\xi +(\gamma -1)J]/(\gamma +1)$, $c=\sigma (\gamma -1)(\xi -J)/(\gamma +1)$, $\rho =(c^2/(\gamma K){)}^{1/(\gamma -1)}$ and $p=K{\rho }^{\gamma }$, on intervals where $c>0$. The specific entropy is constant. Together with uniform branches, these exhaust smooth positive-gas self-similar solutions; a contact discontinuity is not a smooth fan.

Sensible heat is transferred heat associated with a temperature change within a single thermodynamic phase. By contrast, latent heat can change the phase fraction while the temperature stays fixed during equilibrium phase coexistence. Neither name denotes an additional stored substance: both describe how transferred heat participates in a process.

For fixed mass $m$ and composition, use the heat capacity appropriate to the heating constraint. Under the quasistatic process assumptions in heat capacity, with only pressure-volume work and unchanged bulk kinetic energy and potential energy, fixed-pressure heating givesHere $c_p$ is specific heat capacity at constant pressure; fixed-volume heating instead uses specific heat capacity at constant volume $c_V$. A cooling step has negative $Q_{\mathrm{sens}}$ with the sign convention in heat. A temperature difference has the same numerical value in units of the kelvin or degree Celsius. The integral follows by adding the small transfers $\delta Q=m{c}_p(T)dT$, rather than assuming a constant heat capacity.

As a stylized ice-to-water example for a fixed-composition closed system at fixed pressure $1\mathrm{atm}$, take $m=0.5\mathrm{kg}$, an initial temperature of $-10{}^{\circ }\mathrm{C}$, melting at $0{}^{\circ }\mathrm{C}$, and a final temperature of $20{}^{\circ }\mathrm{C}$. Assume constant specific heat capacity at constant pressure values $c_{\mathrm{ice}}=2.1\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and $c_{\mathrm{water}}=4.18\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, and latent heat of fusion $L_f=334\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$. These are stipulated, rounded model values; ignore the container and heat losses, retain all the material, and assume only pressure-volume work under the quasistatic process conditions above. The three steps areOnly the first and third terms are sensible heat; the middle term is latent heat. The plateau begins after $10.5\mathrm{kJ}$ and ends after $177.5\mathrm{kJ}$ of total supplied heat. At $94\mathrm{kJ}$ total input, half the mass is liquid water and the temperature remains $0{}^{\circ }\mathrm{C}$. The equilibrium phase transition temperature depends on pressure; this example stipulates a single fixed pressure. ^[ref]

This split makes heating budgets easy to interpret: melting consumes $167/219.3\approx 76.2\%$ of the supplied heat in this model despite causing no temperature rise. Within either single thermodynamic phase, the heating curve slope is $dT/dQ=1/(m{c}_p)$; a larger heat capacity makes the slope shallower. A flat interval represents changing phase fraction, so it cannot be modeled by applying $Q=m{c}_p\Delta T$ across the phase transition. The complementary constructed example in latent heat tracks that phase fraction explicitly.

A sphere of radius $R$ settling in a stationary cylindrical tube of radius $R+d$, with $d\ll R$ and zero net through-flow, has leading speedThe neck gap is $H(z)=d+z^2/(2R)$. In the sphere frame, leading bypass flux per circumferential length is $q\simeq UR/2$; wall-motion transport is smaller by $d/R$. The pressure-driven flux through a parabolic lubrication gap gives $\Delta p=9\pi \mu UR\sqrt{2R}/(4d^{5/2})$. Pressure dominates the sphere drag, $F\simeq \pi R^2\Delta p$, and balancing $F$ against $4\pi R^3\Delta \rho g/3$ proves the speed. Tube end conditions allowing appreciable net flow change the bypass balance and can give other drag regimes.

A shock polar is the locus of possible downstream velocity vectors, or an equivalent pressure/deflection relation, as the orientation of an oblique shock varies for fixed upstream conditions. Entropy admissibility selects physical compressive shocks. The strong-shock perfect-gas velocity polar is a circle, while finite-Mach polars have a different shape.

For a horizontal field, set $C=B^2/{\mu }_0$. Static force balance gives $\delta \Pi =\rho g{\xi }_z-(\gamma p+C)\nabla \cdot \xi +Ci{k}_y{\xi }_y$. Given smooth compactly supported vertical and along-field displacements, the displayed choice makes total-pressure perturbation zero while ${\xi }_x=O(k_x^{-1})$. The nonnegative pressure and transverse-tension terms vanish as $|k_x|\to \infty$. This constructs the infimum needed for the magnetohydrodynamic energy principle; a strictly negative limit is already negative at some finite wavenumber.

A thermodynamic system for which the reversible mechanical work is entirely pressure-volume work. At fixed composition its internal energy is specified by entropy and volume; additional work modes require additional state variables.

A simple fluid has local stress determined by the history of deformation of a material element, rather than by spatial derivatives of that history. An isotropic incompressible simple fluid has an arbitrary isotropic pressure and an objective constitutive functional of the relative deformation gradients. Internal-variable equations can represent such a functional when the initial material state is fixed by the past history.

Sintering joins particles into a coherent solid by heating, sometimes with applied pressure, without completely melting the material. It is used to consolidate powders of ceramics and metals.

For a gravity-current nose of height $h_N$ on a small incline of slope $\theta$, the travelling front of a gravity-driven thin film satisfies $h_y=\theta (1-h_N^2/h^2)$. Close to its dry tip, $h\sim (-3\theta h_N^{2}y{)}^{1/3}$. The slope becomes order one at $h\sim \sqrt{\theta }{h}_N$, giving an $O(\sqrt{\theta }{h}_N)$ region where lubrication theory fails. The main nose length is $h_N/\theta$. Comparing capillary pressure gradients $\gamma h_N/(h_N/\theta {)}^3$ with gravitational forcing $\rho g\theta$ gives the crossover $h_N\sim \theta \sqrt{\gamma /(\rho g)}$. This concerns the main nose length; capillarity may already matter in a much smaller tip region.